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A Non-Abelian Conjecture of Tate–Shafarevich Type for Hyperbolic

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A Non-Abelian Conjecture of Tate–Shafarevich Type for Hyperbolic A NON-ABELIAN CONJECTURE OF TATE–SHAFAREVICH TYPE FOR HYPERBOLIC CURVES JENNIFER S. BALAKRISHNAN, ISHAI DAN-COHEN, MINHYONG KIM, AND STEFAN WEWERS Abstract. Let X denote a hyperbolic curve over Q and let p denote a prime of good reduction. The third author’s approach to integral points, introduced in [Kim2] and [Kim3], endows X(Zp) with a nested sequence of subsets X(Zp)n which contain X(Z). These sets have been computed in a range of special cases [Kim4, BKK, DCW2, DCW3]; there is good reason to believe them to be practically computable in general. In 2012, the third author announced the conjecture that for n sufficiently large, X(Z)= X(Zp)n. This conjecture may be seen as a sort of compromise between the abelian confines of the BSD conjecture and the profinite world of the Grothendieck section conjecture. After stating the conjecture and explaining its relationship to these other conjectures, we explore a range of special cases in which the new conjecture can be verified. 2010 Mathematics Subject Classification: 11D45, 14H52, 11G50, 14F35 1. Introduction 1.1. When E/Q is an elliptic curve, the conjecture of Birch and Swinnerton- Dyer predicted the following phenomenon: L(E, 1) = 0 E(Q) < . 6 ⇒ | | ∞ This is now a theorem, strikingly realized by the process of annihilating the Mordell-Weil group with the L-value in question [Kol, Kat]. When we move to the realm of hyperbolic curves, that is, curves with non-abelian arXiv:1209.0640v4 [math.NT] 2 Apr 2017 geometric fundamental groups, we have suggested elsewhere an extension of this connection between Diophantine finiteness and non-vanishing of L- values [CK, Kim5], even though it has thus far proved difficult to formulate it in precise terms. 1.2. The goal of this paper is to extend a different part of the constellation of conjectures surrounding BSD, namely, the finiteness of the Tate-Shafarevich group ШE. To explain this, we begin by turning our attention to a different conjecture, namely Grothendieck’s section conjecture. Let X be a compact hyperbolic curve over Q, let X¯ denote the base change of X to an algebraic Date: April 4, 2017. 1 2 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS closure of Q with Galois group G, and let b be a Q-valued point of X. Then according to the conjecture, the map x [πét(X¯; b, x)] 7→ 1 ét ¯ that associates to a rational point x the π1 (X, b)-torsor of paths from b to x defines a bijection X(Q) H1(G, πét(X,¯ b)). ≃ 1 Returning to the special case of an elliptic curve E, our point of departure is the apparent similarity between this bijection and a certain isomorphism implied by the conjectured finiteness of ШE, namely 1 (*) E(Q) Z Q H (G, H (E,¯ Q )). ⊗ p ≃ Z 1 p Here, the subscript ‘Z’ refers to the cohomology classes for the group G that are crystalline at p, and zero at all v = p. 6 1.3. For the conjecture being presented here, we let Spec Z be a regular minimal Z-model of a hyperbolic curve (see 2.1 forX a precise → definition); its generic fiber X = Q need not be proper. We let b be an integral base point (possibly tangential),X and assume p is a prime of good reduction for and b. Between the profinite fundamental group of the section conjecture,X and the first étale homology of segment 1.1, equation (*), lies the unipotent p-adic th étale fundamental group U of XQ¯ at b. We let Un denote its n quotient along the descending central series. Let GQ denote the total Galois group of Q and let Gp denote the total Galois group of Qp. Following [Kim2, Kim3], we consider the subspace H1(G ,U ) H1(G ,U ) f p n ⊂ p n consisting of Gp-equivariant Un-torsors which are crystalline. We also con- sider a certain subspace n 1 Sel ( ) H (GQ,U ), X ⊂ n the Selmer scheme of ; roughly speaking, it parametrizes those torsors X which are crystalline at p and in the image of (Zv) (we say locally geometric) for v = p. For each n these fit into a commutingX square like so, 6 X(Z) / X(Zp) j jp n / 1 Sel ( ) Hf (Gp,Un) X locp and we define 1 n (Z ) := j− loc (Sel ( )) . X p n p p X These form a nested sequence of subsets like so. (Z ) (Z ) (Z ) (Z). X p ⊃X p 1 ⊃X p 2 ⊃···⊃X The conjecture, which was first proposed by M.K. in his lectures at the I.H.E.S. in February of 2012, is as follows. ANON-ABELIANCONJECTURE 3 Conjecture (3.1 below). Equality (Z ) = (Z) is obtained for large n. X p n X n 1.4. We also suggest a variant of the Selmer scheme SelS( ) of , suited 1 X X to computing the Z[S− ]-valued points of for S a finite set of primes, by dropping the local geometricity condition overX S. This gives rise to a square 1 X(Z[S− ]) / X(Zp) jS jp n / 1 SelS( ) Hf (Gp,Un), X locp and to an associated sequence of subsets (Z ) (Z ) (Z ) (Z), X p ⊃X p S,1 ⊃X p S,2 ⊃···⊃X for which equality (Z ) = (Z[S 1]) may hold for large n. X p S,n X − 1.5. These constructions are based on the third author’s approach to inte- gral points, introduced in [Kim2] and [Kim3]. The relationship to the section conjecture has been explored before. For instance in [Kim6], the third au- thor shows that if (Zp)n = (Zp) for some n, then the section conjecture would in principleX allow one6 X to obtain a computable bound on the height of rational points. Our present conjecture, however, is quite different in fla- vor from the section conjecture and its direct consequences. It shares more with the conjectures of Tate–Shafarevich and Birch–Swinnerton-Dyer, both in terms of concreteness and in terms of computability. Indeed, like the BSD conjecture, the present conjecture can actually be tested numerically. 1.6. Our principal goal below is to do just that. Work completed elsewhere allows us to verify our conjecture in a range of cases. New in this article is the case of a punctured elliptic curve of rank zero: we are able to compute the sets (Zp)2, and subsequently to verify the conjecture for many such curves. ThisX computation is based on a study of the unipotent Kummer map j : X(F ) H1(G ,U ) v v → v 2 for a punctured elliptic curve X over a local field Fv of residue characteristic v = p. Our main theorem (4.1.6) says that the p-adic height function can be6 retrieved from this map. This is of interest in its own right, and is suggestive of the possibility of obtaining functions on the local points from higher quotients of the unipotent fundamental group which might play a role similar to the role played by heights here. This point of view is implicit in the third author’s work on nonabelian reciprocity laws [Kim1] and in ongoing joint work between him and Jonathan Pridham. 4 BALAKRISHNAN,DAN-COHEN,KIM,ANDWEWERS 1.7. As explained in [Kim2, Kim3], the key to computing the map jp is its equivalence with a certain p-adic analog α : (Z ) U DR/F 0 X p → n of the higher Albanese map of Richard Hain [Hai] through a lifting of the Bloch-Kato exponential to the unipotent level obtained via the unipotent p-adic Hodge theory of Martin Olsson [Ols]. The p-adic unipotent Albanese map α is given in coordinates by certain p-adic iterated integrals (known also as Coleman functions), and there are fairly well-established methods for producing explicit formulas for the resulting iterated integrals on the one hand, and for computing p-adic approximations of their values on the other. For instance, the case of the thrice punctured line was treated by Furusho [Fur1, Fur2] and by Besser–de Jeu [BdJ]. The problem of explicit determi- nation of the unipotent Albanese map for punctured elliptic curves in depth two is treated by Kim [Kim4] and Balakrishnan–Kedlaya–Kim [BKK]. The problem of computing Coleman functions on hyperelliptic curves is treated by Balakrishnan–Bradshaw–Kedlaya [BBK] and by Balakrishnan [Bal2]. 1.8. We turn to the map loc : Seln( ) H1(G ,U ). p X → f p n This is actually an algebraic map of finite-type affine Qp-schemes; its target is in fact isomorphic to affine space. Let (n) denote the ideal defining its scheme-theoretic image. As explained inL [Kim3], as soon as (n) = 0, L 6 (Zp)n becomes finite. Moreover, several well known motivic conjectures (Fontaine-Mazur-Jannsen,X Bloch-Kato) imply that for n large, j∗ (n + 1) ) j∗ (n), p L p L and, in fact, the larger ideal contains elements that are algebraically inde- 1 pendent of the elements in j∗ (n). So a point in the common zero set for all n should be there for a goodL reason; our conjecture expresses the belief that such a point must belong to (Z). X 1.9. The study of the ideals (n) relates not only the the plausibility of our conjecture, but also to its usefulness.L Explicit computation of these ideals has been achieved in a range of special cases. An approach to the case of the thrice punctured line using the methods of mixed Tate motives is cur- rently under development in a sequence of articles by Dan-Cohen and Wewers [DCW2, DCW3, DC].
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